##### 4.3.25 $$y'(x)=\sec ^2(x) \sec ^3(y(x))$$

ODE
$y'(x)=\sec ^2(x) \sec ^3(y(x))$ ODE Classiﬁcation

[_separable]

Book solution method
Separable ODE, Neither variable missing

Mathematica
cpu = 0.941103 (sec), leaf count = 2959

$\left \{\left \{y(x)\to -\cos ^{-1}\left (-\frac {\sqrt {2^{2/3} \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2+\frac {2 \sqrt [3]{2}}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}}{\sqrt {2}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (-\frac {\sqrt {2^{2/3} \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2+\frac {2 \sqrt [3]{2}}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}}{\sqrt {2}}\right )\right \},\left \{y(x)\to -\cos ^{-1}\left (\frac {\sqrt {2^{2/3} \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2+\frac {2 \sqrt [3]{2}}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}}{\sqrt {2}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (\frac {\sqrt {2^{2/3} \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2+\frac {2 \sqrt [3]{2}}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}}{\sqrt {2}}\right )\right \},\left \{y(x)\to -\cos ^{-1}\left (-\frac {1}{2} \sqrt {-\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}-2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (-\frac {1}{2} \sqrt {-\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}-2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to -\cos ^{-1}\left (\frac {1}{2} \sqrt {-\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}-2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (\frac {1}{2} \sqrt {-\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}-4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}-2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to -\cos ^{-1}\left (-\frac {1}{2} \sqrt {\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}+2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (-\frac {1}{2} \sqrt {\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}+2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to -\cos ^{-1}\left (\frac {1}{2} \sqrt {\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}+2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \},\left \{y(x)\to \cos ^{-1}\left (\frac {1}{2} \sqrt {\frac {i \left (2^{2/3} \sqrt {3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+i 2^{2/3} \left (-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2\right ){}^{2/3}+4 i \sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}-2 \sqrt [3]{2} \sqrt {3}+2 i \sqrt [3]{2}\right )}{\sqrt [3]{-9 c_1{}^2-18 \tan (x) c_1-9 \tan ^2(x)+3 \sqrt {(c_1+\tan (x)){}^2 \left (9 c_1{}^2+18 \tan (x) c_1+9 \tan ^2(x)-4\right )}+2}}}\right )\right \}\right \}$

Maple
cpu = 0.21 (sec), leaf count = 71

$\left [y \left (x \right ) = \arctan \left (\frac {3 \textit {\_C1} +3 \tan \left (x \right )}{\RootOf \left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 \textit {\_C1}^{2}+18 \textit {\_C1} \tan \left (x \right )+9 \left (\tan ^{2}\left (x \right )\right )-4\right )^{2}+2}, \RootOf \left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 \textit {\_C1}^{2}+18 \textit {\_C1} \tan \left (x \right )+9 \left (\tan ^{2}\left (x \right )\right )-4\right )\right )\right ]$ Mathematica raw input

DSolve[y'[x] == Sec[x]^2*Sec[y[x]]^3,y[x],x]

Mathematica raw output

{{y[x] -> -ArcCos[-(Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan
[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])
^(1/3) + 2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Ta
n[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2])]}, {y[x]
 -> ArcCos[-(Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 +
 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)
+ 2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2
*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2])]}, {y[x] -> -Ar
cCos[Sqrt[-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[
(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + 2^(2/3
)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9
*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2]]}, {y[x] -> ArcCos[Sqrt[
-2 + (2*2^(1/3))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Ta
n[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + 2^(2/3)*(2 - 9*C
[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 +
18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/Sqrt[2]]}, {y[x] -> -ArcCos[-1/2*Sqrt[((-I
)*((-2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9
*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^
2)])^(1/3) - I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1
] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sq
rt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4
 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan
[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*
Tan[x]^2)])^(1/3)]]}, {y[x] -> ArcCos[-1/2*Sqrt[((-I)*((-2*I)*2^(1/3) - 2*2^(1/3
)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + T
an[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) - I*2^(2/3)*(2 -
9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2
 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1
]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x]
 + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C
[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]]}, {y[x]
-> -ArcCos[Sqrt[((-I)*((-2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2
- 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1
]*Tan[x] + 9*Tan[x]^2)])^(1/3) - I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Ta
n[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)]
)^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(
C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*
C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 +
 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x] -> ArcCos[Sqrt[((-I)*((-2*I)*2
^(1/3) - 2*2^(1/3)*Sqrt[3] - (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 +
 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)
- I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])
^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 -
9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2
 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan
[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])
^(1/3)]/2]}, {y[x] -> -ArcCos[-1/2*Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] +
(4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4
 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 1
8*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*T
an[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9
*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^
2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x]
)^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]]}, {y[x] -> ArcCos[-1
/2*Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] + (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Ta
n[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9
*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*S
qrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^
(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x
])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*
C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan
[x] + 9*Tan[x]^2)])^(1/3)]]}, {y[x] -> -ArcCos[Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3
)*Sqrt[3] + (4*I)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + T
an[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 -
9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2
 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1
]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x]
 + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C
[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}, {y[x
] -> ArcCos[Sqrt[(I*((2*I)*2^(1/3) - 2*2^(1/3)*Sqrt[3] + (4*I)*(2 - 9*C[1]^2 - 1
8*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*T
an[x] + 9*Tan[x]^2)])^(1/3) + I*2^(2/3)*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x
]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(
2/3) + 2^(2/3)*Sqrt[3]*(2 - 9*C[1]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1
] + Tan[x])^2*(-4 + 9*C[1]^2 + 18*C[1]*Tan[x] + 9*Tan[x]^2)])^(2/3)))/(2 - 9*C[1
]^2 - 18*C[1]*Tan[x] - 9*Tan[x]^2 + 3*Sqrt[(C[1] + Tan[x])^2*(-4 + 9*C[1]^2 + 18
*C[1]*Tan[x] + 9*Tan[x]^2)])^(1/3)]/2]}}

Maple raw input

dsolve(diff(y(x),x) = sec(x)^2*sec(y(x))^3, y(x))

Maple raw output

[y(x) = arctan(3*(_C1+tan(x))/(RootOf(_Z^6+3*_Z^4+9*_C1^2+18*_C1*tan(x)+9*tan(x)
^2-4)^2+2),RootOf(_Z^6+3*_Z^4+9*_C1^2+18*_C1*tan(x)+9*tan(x)^2-4))]